Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 7/refcontrol

Exercises

Exercise Create referencenumber

Have you ever seen in a newspaper an expression of the form $2.\overline{17}$ (a vinculum to indicate a repeating decimal) or $\mathrm{2.17171717...}$? Why not? What kind of decimal expansions occur in a newspaper (in a manual, as a result of a measurement, in a technical description)?

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We want to find a rational approximation for the number $1000000\pi$, such that the deviation of the true value should be at most $\frac{1}{1000}$. How good has to be an approximation for $\pi$ such that we obtain the asked for approximation?

Exercise Create referencenumber

Is there any relation between the Din-Norm for paper and square roots?

Exercise Create referencenumber

Compute by hand the approximations $x_1,x_2,x_3,x_4$ in Heron's method to find the square root of $5$ with initial value $x_0=2$.

Exercise Create referencenumber

Do the first three steps in Heron's method to compute the square root of $b=7$ with initial value $x_0=3$ (so the approximations $x_1,x_2,x_3$ for $\sqrt{7}$ shall be computed; these numbers have to be given as fractions reduced to lowest terms).

Exercise Create referencenumber

Let $c\in {ℝ}_{+}$ be a positive real number and let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be the Heron-sequenceMDLD/Heron-sequence for the computation of $\sqrt{c}$ with the initial value $x_0\in {ℝ}_{+}$. Let $u\in {ℝ}_{+}$, $d=c\cdot u^2$, $y_0=u{x}_0$ and let ${\left(y_n\right)}_{n\in \mathbb{N}}$ be the Heron-sequence for the computation of $\sqrt{d}$ with initial value $y_0$. Show that

$y_n=u{x}_n$

holds for all $n\in \mathbb{N}$.

Exercise Create referencenumber

Determine for the sequence

$x_n:=\frac{2}{3n+5}$

and

$ϵ=\frac{1}{10},\frac{1}{100},\frac{1}{1000},\frac{1}{10000},\dots ,$

for which (minimal) $n$ the estimate

$x_n\le ϵ$

holds.

===Exercise Exercise 7.7

change===

Let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a real sequence. Prove that the sequence converges to $x$ if and only if for all $k\in {\mathbb{N}}_{+}$ a natural number $n_0\in \mathbb{N}$ exists, such that for all $n\ge n_0$ the estimation $|x_n-x|\le \frac{1}{k}$ holds.

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Negate the statement that a sequence $x_n$ convergesMDLD/converges (R) in $ℝ$ to $x$ by transforming the quantifiers.

Exercise Create referencenumber

Examine the convergence of the following sequence

$x_n=\frac{1}{n^2}$

where $n\ge 1$.

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Regarding the sequence $x_n:=\frac{n}{2n}$, somebody says: "The numerator and the denominator are both going to infinity. However, the denominator is much faster, therefore the sequence converges to $0$“. What do you think about this argument?

Exercise Create referencenumber

Determine whether the following subsets $M\subseteq ℝ$ are boundedMDLD/bounded (R) or not.

1. $\mathbb{N}$,
2. $\{\frac{1}{2},\frac{-3}{7},\frac{-4}{9},\frac{5}{9},\frac{6}{13},\frac{-1}{3},\frac{1}{4}\}$,
3. $]-5,2]$,
4. $\{\frac{1}{n}\mid n\in {\mathbb{N}}_{+}\}$,
5. $\{\frac{1}{n}\mid n\in {\mathbb{N}}_{+}\}\cup \{0\}$,
6. ${\mathbb{Q}}_{-}$,
7. $\{x\in \mathbb{Q}\mid x^2\le 2\}$,
8. $\{x\in \mathbb{Q}\mid x^2\ge 4\}$,
9. $\{x^2\mid x\in \mathbb{Z}\}$.

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Let $x>1$ be a real number.MDLD/real number Show that the sequence $x_n:=x^n$ is not bounded.MDLD/bounded (R)

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Let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a null sequence and let ${\left(y_n\right)}_{n\in \mathbb{N}}$ be a bounded real sequence. Show that also the product sequence $(x_n{y}_n{)}_{n\in \mathbb{N}}$ is a null sequence.

===Exercise Exercise 7.16

change===

Let ${\left(x_n\right)}_{n\in \mathbb{N}}$ and ${\left(y_n\right)}_{n\in \mathbb{N}}$ be convergent real sequences with $x_n\ge y_n$ for all $n\in \mathbb{N}$. Prove that $\underset{n\to \mathrm{\infty }}{lim}{x}_n\ge \underset{n\to \mathrm{\infty }}{lim}{y}_n$ holds.

===Exercise Exercise 7.17

change===

Let ${\left(x_n\right)}_{n\in \mathbb{N}},{\left(y_n\right)}_{n\in \mathbb{N}}$ and ${\left(z_n\right)}_{n\in \mathbb{N}}$ be three real sequences. Let $x_n\le y_n\le z_n$ for all $n\in \mathbb{N}$ and ${\left(x_n\right)}_{n\in \mathbb{N}}$ and ${\left(z_n\right)}_{n\in \mathbb{N}}$ be convergent to the same limit $a$. Prove that also ${\left(y_n\right)}_{n\in \mathbb{N}}$ converges to the same limit $a$.

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Let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a convergent sequence of real numbers with limit equal to $x$. Prove that also the sequence

${(|x_n|)}_{n\in \mathbb{N}}$

converges, and specifically to $|x|$.

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Let the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ be given by

$x_n=\{\begin{array}{c}1,\text{ if }n\text{ is a prime number },\\ 0\text{ else}.\end{array}$

1. Determine $x_{117}$ and $x_{127}$.
2. Does this sequence converge in $\mathbb{Q}$?

Exercise Create referencenumber

Prove by induction the Binet formula for the Fibonacci numbers. This says that

$f_n=\frac{{\left(\frac{1+\sqrt{5}}{2}\right)}^n-{\left(\frac{1-\sqrt{5}}{2}\right)}^n}{\sqrt{5}}$

holds ($n\ge 1$).

Hand-in-exercises

===Exercise (3 marks) Create referencenumber=== Compute by hand the approximations $x_1,x_2,x_3,x_4$ in Heron's method to find the square root of $7$ with initial value $x_0=2$.

Exercise (5 marks) Create referencenumber

Write a computer-program (pseudocode) for the computation of rational approximations for the square root of a rational number using Heron's method.MDLD/Heron's method

• The computer has as many memory units as needed, which can contain natural numbers.
• It can compare the content of memory units and can, depending on the outcome, switch to a certain program line.
• It can add the content of two memory units and write the result into another memory unit.
• It can multiply the content of two memory units and write the result into another memory unit.
• It can print contents of memory units and it can print given texts.
• There is a stop command.

The initial configuration is

$(a,b,c,d,e,0,0,\dots )$

with $b,c,e\ne 0$. Here, $a/b$ is the number from which we want to compute the square root, $x_0=c$ is the initial value and $d/e$ is the wished-for accuracy. The program shall compute and print the Heron-sequence $x_0,x_1,x_2,\dots$ (the numerators and denominators are printed successively) and it shall stop when the member $x_n$ printed last fulfills the property

$|x_n^2-\frac{a}{b}|\le \frac{d}{e}.$

Attention! All operations are to be done within $\mathbb{N}$!

Exercise (3 marks) Create referencenumber

Determine for the sequence

$x_n:=\frac{2n+1}{3n-4}$

and for

$ϵ=\frac{1}{10},\frac{1}{100},\frac{1}{1000},$

for which (minimal) $n$ the estimate

$|x_n-\frac{2}{3}|\le ϵ$

holds.

Exercise (6 marks) Create referencenumber

Let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a convergentMDLD/convergent (R) real sequenceMDLD/real sequence with limitMDLD/limit (real sequence) $x$. Show that the sequence defined by

$y_n:=\frac{x_0+x_1+\cdots +x_n}{n+1}$

also converges to $x$.
Hint: reduce to the case $x=0$.

Exercise (4 marks) Create referencenumber

Prove that the real sequence

$\frac{n}{2^n}$

converges to $0$.
Hint: Find a suitable estimate for $2^n$ using the binomial theorem.

Exercise (5 marks) Create referencenumber

Let $(x_n{)}_{n\in \mathbb{N}}$ and $(y_n{)}_{n\in \mathbb{N}}$ be sequences of real numbers and let the sequence $(z_n{)}_{n\in \mathbb{N}}$ be defined as $z_{2n-1}:=x_n$ and $z_{2n}:=y_n$. Prove that $(z_n{)}_{n\in \mathbb{N}}$ converges if and only if $(x_n{)}_{n\in \mathbb{N}}$ and $(y_n{)}_{n\in \mathbb{N}}$ converge to the same limit.

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